Difference between revisions of "Lab Manual: Limits of Detection"

Revision as of 03:26, 25 November 2012

20.309: Biological Instrumentation and Measurement

Second order circuit with alternating current source and parallel resistor, capacitor, and inductor.

$ \dfrac{1}{Z_{eq}}=\dfrac{1}{Z_R}+\dfrac{1}{Z_L}+\dfrac{1}{Z_C} $

$ Z_{eq}=\frac{Z_R Z_L Z_C}{Z_R Z_L + Z_R Z_C + Z_L Z_C} $

$ Z_{eq}=\frac{\hat{V}_o(s)}{\hat{I}_{in}(s)}=\frac{RL/C}{RLs+\dfrac{R}{Cs}+\dfrac{L}{C}}=\frac{Ls}{LCs^2+\dfrac{L}{R}s+1} $

Ideal mechanical and electronic lumped elements.

$ \frac{\hat{V}_o(s)}{\hat{F}_{in}(s)}=\frac{\dfrac{1}{kb}s}{\dfrac{m}{k} s^2+ \dfrac{m}{b} s+1} $

@@ Line 11: / Line 11: @@
 ===Second order system===
-[[Image:Second Order Circuit.png|thumb|300 px|right|Second order circuit with alternating current source and parallel resistor, capacitor, and inductor.]]
+[[Image:Second Order Circuit.png|thumb|right|Second order circuit with alternating current source and parallel resistor, capacitor, and inductor.]]
-<math>Z_{eq}=\frac{\hat{V}_o(s)}{\hat{I}_{in}(s)}=\frac{Z_R Z_L Z_C}{Z_R+Z_L+Z_C}=\frac{RL/C}{R+Ls+1/Cs}</math>
+::<math>\dfrac{1}{Z_{eq}}=\dfrac{1}{Z_R}+\dfrac{1}{Z_L}+\dfrac{1}{Z_C}</math>
-<math>\frac{\hat{V}_o(s)}{\hat{I}_{in}(s)}=\frac{RLs}{LCs^2+RCs+1}</math>
+::<math>Z_{eq}=\frac{Z_R Z_L Z_C}{Z_R Z_L + Z_R Z_C + Z_L Z_C}</math>
+::<math>Z_{eq}=\frac{\hat{V}_o(s)}{\hat{I}_{in}(s)}=\frac{RL/C}{RLs+\dfrac{R}{Cs}+\dfrac{L}{C}}=\frac{Ls}{LCs^2+\dfrac{L}{R}s+1}</math>
 ===Mechanical circuit analogy===
-[[Image:Ideal Mechanical and Electronic Lumped Elements.png|thumb|right|350 px|Ideal mechanical and electronic lumped elements.]]
+[[Image:Ideal Mechanical and Electronic Lumped Elements.png|thumb|right|Ideal mechanical and electronic lumped elements.]]
+<math>\frac{\hat{V}_o(s)}{\hat{F}_{in}(s)}=\frac{\dfrac{1}{kb}s}{\dfrac{m}{k} s^2+ \dfrac{m}{b} s+1}</math>
 ===Underdamped system: atomic force microscope===